MACHINE THEORY
Bachelor in Mechanical Engineering
CAMS DESIGN
Ignacio Valiente Blanco
José Luis Pérez Díaz
David Mauricio Alba Lucero
Efrén Díez Jiménez
INDEX
1. Introduction
2. Cams clasification
3. Desmodromic Cams
4. Followers clasification
5. Closure Joints clasification
6. Classification of Cams mechanisms
7. SVAJ Diagrams
8. Fundamental Law of Cams
► 8.1 Simple Harmonic Motion
► 8.2 Polynomial Functions
► 8.3 Functions Comparison
9. What can cams do?
Literature
2
1. Introduction
Cam mechanism – a very versatile mechanical element for
transforming an input motion into another. It is mainly
composed of two parts:
Cam: Input linckage. Motion is usually rotation.
Follower: Output linckage. Translation or oscillation.
3
1. Introduction
Advantages of cams
•
•
•
•
Many possibilities of motion transformation. Very versatile.
Have few parts.
Take small space.
Widely used in industry. Known technology.
Limitations of cams
•
•
•
•
Suceptible to vibrations.
Wear.
Fatigue.
Lubrication needed.
4
1. Introduction
Cams are usually classified according to the cam geometry:
► Planar
cams
Disc or plate cams
Linear cams: Cam do not rotate. Backwards and forwards motion.
5
2. Cams clasification
►
Spatial cams (three-dimensional)
Cylindrical cams: The roller follower operates in a groove cut on the
periphery of a cylinder. The follower may translate or oscillate
Conical cams:
6
2. Cams clasification
7
2. Cams clasification
Globoidal cams: compact structrure, high load capacity, low noise and
vibrations and high reliability. Widely used in industry.
Spherical end cams
8
2. Cams clasification
Cams can be classified according to the follower motion respect
the rotation axle of the cam
►
Radial Cams
► Translational
or Axial Cams
9
3. Desmodromic Cams
Desmodromic cams use two higher pairs (closed form) so that
both impose the same kinematic restrictions.
There are four types of planar desmodromic cams:
Cams with
constant width
Cams with
constant diameter
(double flat faced
follower)
(double roller follower)
Conjugate
cams with
double follower
Cams with
a grooved
face
10
4. Followers clasification
Followers can be classified by the geometry of the contact
surface. E.g.
►
Flat-face: More compact and less
► Roller: Less susceptible to wear.
expensive than roller. Widely use in
automotive industry
Easy to replace the roller with a short
delay. Widely used in production
machinery
►
Knife edge follower and others
11
4. Followers clasification
Followers can also be classified according to their motion:
► Translational
► Oscillating
12
5. Closure Joints clasification
Joint closure between the cam and the follower is an important point to
take into account when design a cam mechanism. It is a high kinematic
pair (See Introduction to kinematics and mechanisms)
► Force
Joints: An external force assure permanent contact between the follower
and the cam surface. Springs, gravity and inertial systems.
13
5. Closure Joints clasification
► Form
joints: An appropriated form of a groove in the cam provides the path
to be covered by the follower. No external force is needed.
14
6. Classification of Cam
mechanisms
Cam mechanisms can be classified according to motion
restrictions:
► CEP: Critical extreme position. Only define start and stop position of the
follower.
► CPM: Critical path motion. Path and/or one or more of its derivates are
carefully specified.
Cams can be classified according to their motion program:
► RF:
Rise-Fall. CEP (No Dwell)
► RFD: Rise-Fall-Dwell.CEP (Single Dwell)
► RDFD: Rise-Dwell-Fall-Dwell. CEP (Double Dwell)
15
7. SVAJ Diagrams
•
•
The first step when design cam is to define the mathematical
functions to be used to define the motion of the follower.
SVAJ diagrams are a useful tool:
S: displacement of the follower vs θ of the cam.
V: velocity.
A: acceleration.
J: Jerk. (golpe)
s
∂s/ ∂t
∂2s/ ∂t2
∂3s/ ∂t3
16
7. SVAJ Diagrams
E.g.:
A cam mechanism will be designed for a
driller machine.
Input part
Operation conditions:
One part each 20s. Initial position: A
DRILL IN: 25 mm in 5 s. B-C
DWELL
DWELL: 5 s finishing the drill.C-D
DRILL OUT: 25 mm in 5s.D-A
DWELL: Wait for another part.A-B 5s
•
•
•
A-B
B-C
Vdrill
C-D
D-A
¿? QUESTION ¿? What is the motion program? How many dwells are there?
-
17
7. SVAJ Diagrams
E.g.:
A cam mechanism will be designed for a
driller machine.
Input part
Operation conditions:
One part each 20s. Initial position: A
DRILL IN: 25 mm in 5 s. B-C
DWELL
DWELL: 5 s finishing the drill.C-D
DRILL OUT: 25 mm in 5s.D-A
DWELL: Wait for another part.A-B 5s
•
•
•
A-B
B-C
Vdrill
C-D
D-A
¿? QUESTION ¿? What is the motion program? How many dwells are there?
RDFD: Rise-Dwell-Fall-Dwell. 2Dwells
-
18
7. SVAJ Diagrams
First step:
Define the displacement diagram:
A-B
s diagram
25
Displacement of the follower (s)
[mm]
•
C
Vdrill
Input part
D
B-C
20
DWELL
15
C-D
D-A
10
5
A
B
A
0
0
90
180
270
360
θ cam [º]
19
7. SVAJ Diagrams
It is very useful to divide the displacement function in several parts. This functions are called:
piecewise functions:
From A to B
s diagram
Displacement of the follower
(s) [mm]
25
C
D
From B to C
20
15
From C to D
10
From D to A
5
A
B
A
0
0
90
180
270
360
θ cam [º]
20
7. SVAJ Diagrams
A-B
Vdrill
Input part
DWELL
B-C
D-A
C-D
Velocity= ∂s/ ∂t or ∂s/ ∂θ
ωleva = ∆θ/∆t
ωleva =18[º/s]=3 r.p.m constant.
V= ∆s/ ∆t = ωleva—∆s/ ∆θ= ωleva—Kn=constant
K1= 5/18 [mm/º]
K2=-5/18 [mm/º]
V1= 5 mm/s
V2= -5 mm/s
VELOCITY IS USUALLY EXPRESSED IN FUNCTION OF THE
ROTATED ANGLE OF THE CAM [lenght/angle]. In this
21
example it qould be v= 5/18 [mm/º]
7. SVAJ Diagrams
a= ∂v/ ∂t or ∂v/ ∂θ
∞ a, ∞ F?¿¡¡¡¡¡¡¡¡¡¡. Dirac Delta
function. SPIKES
22
7. SVAJ Diagrams
j= ∂a/ ∂t or ∂a/ ∂θ
CONCLUSION= BAD DESIGN OF
DISPLACEMENT FUNCTION
23
8. Fundamental Law of Cams
Design
1.
2.
“The cam function must be continuous through the first and second
derivatives of displacement across the entire interval”
“The jerk function must be finite across the entire interval”
SOME CONSIDERATIONS
1.
In any but the simplest of cams, the cam motion program cannot be defined by a single
mathematical expression, but rather must be defined by several separate functions, each
of which defines the follower behavior over one segment, or piece, of the cam. These
expressions are sometimes called piecewise
2.
3.
functions.
These functions must have third-order continuity (the function plus two derivatives) at all
boundaries.
The displacement, velocity and acceleration functions must have no discontinuities in
them.
24
8. Fundamental Law of Cams
Design
A-B
Vdrill
Input part
DWELL
B-C
C-D
D-A
The example analyzed before does not comply with the fundamental law of cam design¡¡¡¡
25
8.1 Simple Harmonic Motion
The simple harmonic motion is a mathematical function that remain continuous throughout any
number of diferentiations.
A-B
RISE 0≤θ<90º:
Vdrill
Input part
DWELL
B-C
C-D
D-A
Where:
• h is the total stroke of the follower.
• θ is the camshaft angle in each moment.
• β is the maximun angle in this stage.
Footnote θ/ β is a dimensionless parameter that oscillate
between 0 and 1.
26
8.1 Simple Harmonic Motion
S= OK; V= OK, A is not continuous, infinite jerk spikes¡¡¡¡ Is not a good design yet
Despite this deficiency, this profile is popular for cam design in low-speed applications
because it is easy to manufacture
27
8.2 Polynomial Functions
This class of motion function is one of the more versatile types
that can be used for cam design. They are not limited by the
number of dwells. The general form of a polynomial function is:
Where x is the independent variable (θ/β).
The degree of the polynomial is defined by the number of boundary
conditions minus one (because constant C0).
The constant coefficients must be determined for an specific design.
28
8.2 Polynomial Functions
In our example there are:
► We
divide in fall and rise.
RISE
A-B
FALL
Vdrill
Input part
DWELL
B-C
C-D
θ = 0º → s = 0; v = 0; a = 0
θ = 90º → s = h; v = 0; a = 0
D-A
θ = (θc − θi ) = 0º → s = h; v = 0; a = 0
θ = (θc − θi ) = (270 − 180) = 90º → s = 0; v = 0; a = 0
There are six boundary conditions for each stage.
Then, a 5 degree polynomial equation is needed.
29
8.2 Polynomial Functions
RISE
θ = 0 º → s = 0; v = 0; a = 0
θ = 90 º → s = h ; v = 0; a = 0
2
3
Constant
4
θ 
θ 
θ 
θ 
θ
s = Co + C1 + C2   + C3   + C4   + C5  
β
β 
β 
β 
β 
5
2
3
4
θ 
θ 
θ 
θ  
1
v = C1 + 2C2   + 3C3   + 4C4   + 5C5   
β 
β 
β 
β 
 β  
2
3
θ 
θ 
θ  
1 
a = 2  2C2 + 6C3   + 12C4   + 20C5   
β 
β 
β 
 β  
2
θ 
θ  
1 
j = 3  6C3 + 24C4   + 60C5   
β 
β 
 β  
C0
C1
C2
C3
C4
C5
Condition
θ0; s=h
θ0; v=0
θ0; a=0
θ90; eq.
θ90; eq.
θ90; eq.
Evaluation
h
0
0
10h =250
-15h =-375
6h=150
θ = β → s = 0 = C3 + C4 + C5
θ = β → v = 0 = 3C3 + 4C4 + 5C5 →→→→ C3 = 10h; C4 = −15h; C5 = 6h
θ = β → a = 0 = 6C3 + 12C4 + 20C5
30
8.2 Polynomial Functions
RISE
θ = 0 º → s = 0; v = 0; a = 0
θ = 90 º → s = h ; v = 0; a = 0
2
3
Constant
4
θ 
θ 
θ 
θ 
θ
s = Co + C1 + C2   + C3   + C4   + C5  
β
β 
β 
β 
β 
5
2
3
4
θ 
θ 
θ 
θ  
1
v = C1 + 2C2   + 3C3   + 4C4   + 5C5   
β 
β 
β 
β 
 β  
2
3
θ 
θ 
θ  
1 
a = 2  2C2 + 6C3   + 12C4   + 20C5   
β 
β 
β 
 β  
2
θ 
θ  
1 
j = 3  6C3 + 24C4   + 60C5   
β 
β 
 β  
C0
C1
C2
C3
C4
C5
Condition
θ0; s=h
θ0; v=0
θ0; a=0
θ90; eq.
θ90; eq.
θ90; eq.
Evaluation
h
0
0
10h =250
-15h =-375
6h=150
THIS POLYNOMIAL FUNCTION IS CALLED 3-4-5 POLYNOMIAL
MOTION.
31
8.2 Polynomial Functions
C3 θ90; eq. 10h=250
C4 θ90; eq. -15h=375
C5 θ90; eq. 6h=150
32
8.2 Polynomial Functions
FALL
θ = (θ c − θi ) = 0º → s = h; v = 0; a = 0
θ = (θ c − θi ) = (270 − 180) = 90º → s = 0; v = 0; a = 0
2
3
4
θ 
θ 
θ 
θ 
θ
s = Co + C1 + C2   + C3   + C4   + C5  
β
β 
β 
β 
β 
Constant
5
2
3
4
θ 
θ 
θ 
θ  
1
v = C1 + 2C2   + 3C3   + 4C4   + 5C5   
β 
β 
β 
β 
 β  
2
3
θ 
θ 
θ  
1 
a = 2  2C2 + 6C3   + 12C4   + 20C5   
β 
β 
β 
 β  
2
θ 
θ  
1 
j = 3  6C3 + 24C4   + 60C5   
β 
β 
 β  
C0
C1
C2
C3
C4
C5
Condition
θ0; s=h
θ0; v=0
θ0; a=0
θ90; eq.
θ90; eq.
θ90; eq.
Evaluation
h
0
0
-10h =-250
15h =375
-6h=-150
θ = β → s = 0 = h + C3 + C4 + C5
θ = β → v = 0 = 3C3 + 4C4 + 5C5 →→→→ C3 = −10h; C4 = 15h; C5 = −6h
θ = β → a = 0 = 6C3 + 12C4 + 20C5
Footnote: Please notice that C factors are the opposite to the RISE stage
33
8.2 Polynomial Functions
Design is acceptable. No discontinuities and jerk is finite for any
camshaft angle (fundamental satisfied). However, jerk is “out of control”!
34
8.2 Polynomial Functions
The 4-5-6-7 POLYNOMIAL MOTION
► We
can control jerk by adding two more restrictions and therefore
avoid the initial jerk (θ=0º;j>0) which could be harmful:
Fall
θ = 0º → s = 0; v = 0; a = 0; j = 0
θ = 90º → s = h; v = 0; a = 0; j = 0
Rise
θ = (θ c − θi ) = 0º → s = h; v = 0; a = 0; j = 0
θ = (θ c − θi ) = (270 − 180) = 90º → s = 0; v = 0; a = 0; j = 0
35
8.2 Polynomial Functions
POLYNOMIAL 4-5-6-7 JERK CONTROL
2
3
4
5
6
θ 
θ 
θ 
θ 
θ 
θ 
θ
s = Co + C1 + C2   + C3   + C4   + C5   + C6   + C7  
β
β
β 
β 
β 
β 
β 
7
2
3
4
5
6
θ 
θ 
θ 
θ 
θ 
θ  
1
C1 + 2C2   + 3C3   + 4C4   + 5C5   + 6C6   + 7C7   
β 
β 
β
β 
β 
β 
 β  
2
3
4
5
θ 
θ 
θ 
θ 
θ  
1 
a = 2  2C2 + 6C3   + 12C4   + 20C5   + 30C6   + 42C7   
β 
β
β 
β 
β 
 β  
v=
j=
θ
θ
θ
θ
θ
1 
6C3 + 24C4 
3
β 
β
2
3
4

θ 
θ 
θ  
+
60
C
+
120
C
+
210
C
5
6
7



 

β 
β 
 β  
= β → s = 0 = C4 + C5 + C6 + C7
= β → v = 0 = 4C4 + 5C5 + 6C6 + 7C7
= β → a = 0 = 12C4 + 20C5 + 30C6 + 42C7
= β → j = 0 = 24C4 + 60C5 + 120C6 + 210C7
Constant
Condition
Evaluation
C0
C1
C2
C3
C4
C5
C6
C7
θ0; s=0
θ0; v=0
θ0; a=0
θ0; j=0
θ90; eq.
θ90; eq.
θ90; eq.
θ90; eq.
0
0
0
0
-35h
-84h
70h
-20h
36
8.2 Polynomial Functions
Comparing both polynomial methods:
P 3-4-5 and P 4-5-6-7
1. The follower position is
the same.
2. Maximum
and
minimum velocities are
lower in P 3-4-5,
therefore, the kinetic
energy is also lower.
3. Max
and
min
acceleration are lower
P 4-5-6-7.
4. Initial jerk=0 P 4-5-6-7.
Jerk
function
37
continuous.
8.2 Polynomial Functions
There are many other motion functions which can be used
(see literature).
► The
best function motion will depend on the concrete application:
Single Dwell
Double Dwell
Low-speed applications
High-speed applications
Displacement, velocity or acceleration specifications.
Some examples are: cycloidal displacement, modified trapezoidal
accelerations, modified sinusoidal acceleration….
38
9. SC Levas
We will provide you SC levas. It is a cam design,
simulation and evaluation software.
Very useful tool. Many motion functions available.
Extremely easy to use. Very intuitive
We will use it again in laboratory sessions.
39
10 What can do cams?
http://www.youtube.com/watch?v=2Ah9Mhd6CRA
http://www.youtube.com/watch?v=ykx1DettTDM
http://www.youtube.com/watch?v=ocfIYUc5bpU
http://www.youtube.com/watch?v=ttZJQldd6qM
http://www.youtube.com/watch?v=ESRsd1TeOSg&feature=related
http://www.youtube.com/watch?v=MgklyL0wkVM
http://www.youtube.com/watch?v=0dDiXJsmDOQ
40
Literature
Erdman, A.G., Sandor, G.N. and Sridar Kota Mechanism Design. Prentice Hall,
2001. Fourth Edition.
Robert L. Norton. Design of Machinery. Ed.Mc Graw Hill 1995. Second Edition
Shigley, J.E. & Uicker, J.J. Teoría de máquinas y mecanismos. McGraw-Hill, 1998
SOFTWARE
►
SC levas
VIDEOS
►
http://www.youtube.com/watch?v=2Ah9Mhd6CRA
►
http://www.youtube.com/watch?v=ykx1DettTDM
►
http://www.youtube.com/watch?v=ocfIYUc5bpU
►
http://www.youtube.com/watch?v=ttZJQldd6qM
►
http://www.youtube.com/watch?v=ESRsd1TeOSg&feature=related
►
http://www.youtube.com/watch?v=MgklyL0wkVM
►
http://www.youtube.com/watch?v=0dDiXJsmDOQ
41